Answer: \( A = \frac{c^2}{4\pi} \)
Explanation: The formula provided relates the area \( A \) of a circle to its circumference \( c \). The circumference of a circle is given by \( c = 2\pi r \), where \( r \) is the radius. The area of a circle can also be expressed as \( A = \pi r^2 \). By substituting the expression for \( r \) in terms of \( c \), we can derive the formula for the area from the circumference.
Steps:
- Start with the formula for circumference:
\[
c = 2\pi r
\]
- Solve for the radius \( r \):
\[
r = \frac{c}{2\pi}
\]
- Substitute \( r \) into the area formula \( A = \pi r^2 \):
\[
A = \pi \left(\frac{c}{2\pi}\right)^2
\]
- Simplify the expression:
\[
A = \pi \cdot \frac{c^2}{4\pi^2} = \frac{c^2}{4\pi}
\]
- Thus, the area \( A \) in terms of the circumference \( c \) is:
\[
A = \frac{c^2}{4\pi}
\]
Key Concept: This derivation shows how the area of a circle can be calculated using its circumference, illustrating the relationship between linear and area measurements in geometry.